一句话：用对称幺半闭范畴统一推广句法幺半群，统一句法代数变体，并联系最小自动机与正则语言。
本文针对句法幺半群及其多种变体（有序幺半群、句法半环、结合代数等）缺乏统一范畴论框架的问题，将语言 L 的句法幺半群推广到对称幺半闭范畴 D 中，定义句法 D-幺半群。在 D 为交换代数簇或有序代数簇的假设下，作者证明 L 的句法 D-幺半群可表示为自由 D-幺半群模 L 的句法同余之商，并同构于 L 在 D 中最小自动机的转移 D-幺半群。进一步，当 D 局部有限时，正则语言恰好是句法 D-幺半群有限的语言。该框架统一了 Rabin-Scott、Pin、Polák 与 Reutenauer 等人的句法代数构造。
The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category $\mathcal D$. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott ($\mathcal D=$ sets), the syntactic ordered monoids of Pin ($\mathcal D =$ posets), the syntactic semirings of Pol\'ak ($\mathcal D=$ semilattices), and the syntactic associative algebras of Reutenauer ($\mathcal D$ = vector spaces). Assuming that $\mathcal D$ is a commutative variety of algebras or ordered algebras, we prove that the syntactic $\mathcal D$-monoid of a language $L$ can be constructed as a quotient of a free $\mathcal D$-monoid modulo the syntactic congruence of $L$, and that it is isomorphic to the transition $\mathcal D$-monoid of the minimal automaton for $L$ in $\mathcal D$. Furthermore, in the case where the variety $\mathcal D$ is locally finite, we characterize the regular languages as precisely the languages with finite syntactic $\mathcal D$-monoids.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02694
@article{paperbot406,
title = {A Categorical Approach to Syntactic Monoids},
author = {Jiří Adamek and Stefan Milius and Henning Urbat},
journal = {Logical Methods in Computer Science},
volume = {Volume 14, Issue 2},
year = {2018},
doi = {10.23638/lmcs-14(2:9)2018}
}